There are a lot of papers who can reasonably make that claim (e.g. The Times here in England). The problem, structually, is that newspapers in general are retailers of wholesale product provided by PA, AFP, Reuters, etc etc. Their value-add is in their original content, and that tends to be quite localised i.e. it doesn't scale. I worked in Manhattan for a while, I still have friends there, I enjoy reading that content in the NYT but I've no plans to go back and I wouldn't pay for that content (tho' I will see ads which I will ignore). Original content is expensive to produce. That is the problem for newspapers.
Even if it ran at a loss it still has value due to its prestige. Newscorp recently bought the WSJ for 6B when it was in a similar predicament.